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Mat. Zametki, 1996, Volume 59, Issue 6, Pages 865–880 (Mi mz1785)  

This article is cited in 38 scientific papers (total in 39 papers)

A few remarks on $\zeta(3)$

Yu. V. Nesterenko

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: A new proof of the irrationality of the number $\zeta(3)$ is proposed. A new decomposition of this number into a continued fraction is found. Recurrence relations are proved for some sequences of Meyer's $G$-functions that define a sequence of rational approximations to $\zeta(3)$ at the point 1.


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English version:
Mathematical Notes, 1996, 59:6, 625–636

Bibliographic databases:

UDC: 511.36
Received: 29.11.1995

Citation: Yu. V. Nesterenko, “A few remarks on $\zeta(3)$”, Mat. Zametki, 59:6 (1996), 865–880; Math. Notes, 59:6 (1996), 625–636

Citation in format AMSBIB
\by Yu.~V.~Nesterenko
\paper A~few remarks on $\zeta(3)$
\jour Mat. Zametki
\yr 1996
\vol 59
\issue 6
\pages 865--880
\jour Math. Notes
\yr 1996
\vol 59
\issue 6
\pages 625--636

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    This publication is cited in the following articles:
    1. Sorokin, VN, “On Apery theorem”, Vestnik Moskovskogo Universiteta Seriya 1 Matematika Mekhanika, 1998, no. 3, 48  mathscinet  zmath  isi
    2. Hata, M, “A new irrationality measure for zeta(3)”, Acta Arithmetica, 92:1 (2000), 47  crossref  mathscinet  zmath  isi
    3. W. V. Zudilin, “On the irrationality of the values of the zeta function at odd integer points”, Russian Math. Surveys, 56:2 (2001), 423–424  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    4. Ball, K, “irrationality of an infinite number of values for the zeta function with odd integers”, Inventiones Mathematicae, 146:1 (2001), 193  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. W. V. Zudilin, “Irrationality of values of the Riemann zeta function”, Izv. Math., 66:3 (2002), 489–542  mathnet  crossref  crossref  mathscinet  zmath  elib
    6. W. V. Zudilin, “Diophantine Problems for $q$-Zeta Values”, Math. Notes, 72:6 (2002), 858–862  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. V. N. Sorokin, “Cyclic graphs and Apéry's theorem”, Russian Math. Surveys, 57:3 (2002), 535–571  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. Rivoal, T, “Irrationality of a least one of the nine numbers zeta(5), zeta(7),...,zeta(21)”, Acta Arithmetica, 103:2 (2002), 157  crossref  mathscinet  zmath  adsnasa  isi  scopus
    9. Fischler, S, “Pade approximants and equilibrated hypergeometric series”, Journal de Mathematiques Pures et Appliquees, 82:10 (2003), 1369  crossref  mathscinet  zmath  isi  scopus
    10. Sondow, J, “Criteria for irrationality of Euler's constant”, Proceedings of the American Mathematical Society, 131:11 (2003), 3335  crossref  mathscinet  zmath  isi  scopus
    11. Fischler, S, “Irrationality of zeta values”, Asterisque, 2004, no. 294, 27  mathscinet  zmath  isi
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    13. W. V. Zudilin, “An elementary proof of the irrationality of Tschakaloff series”, J. Math. Sci., 146:2 (2007), 5669–5673  mathnet  crossref  mathscinet  zmath
    14. Zudilin, W, “Well-poised generation of Apery-like recursions”, Journal of Computational and Applied Mathematics, 178:1–2 (2005), 513  crossref  mathscinet  zmath  adsnasa  isi  scopus
    15. T. G. Hessami Pilehrood, Kh. Hessami Pilehrood, “Irrationality of the sums of zeta values”, Math. Notes, 79:4 (2006), 561–571  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    16. Fischler, S, “An exponent of rational approximation of density”, International Mathematics Research Notices, 2006, 95418  mathscinet  zmath  isi  elib
    17. Huttner, M, “Constructible sets of linear differential equations and effective rational approximations of polylogarithmic functions”, Israel Journal of Mathematics, 153 (2006), 1  crossref  mathscinet  zmath  isi  scopus
    18. Rivoal, T, “Euler Numbers, Pade approximants and Catalan's constant”, Ramanujan Journal, 11:2 (2006), 199  crossref  mathscinet  zmath  isi  scopus
    19. Krattenthaler, C, “Basic hypergeometric series, q-analogues of the values of the function zeta and the Eisenstein series”, Journal of the Institute of Mathematics of Jussieu, 5:1 (2006), 53  crossref  mathscinet  zmath  isi  scopus
    20. A. I. Aptekarev, “Preface”, Proc. Steklov Inst. Math., 272, suppl. 2 (2011), S138–S141  mathnet  crossref  crossref  zmath  isi
    21. Elsner, C, “Algebraic relations for reciprocal sums of Fibonacci numbers”, Acta Arithmetica, 130:1 (2007), 37  crossref  mathscinet  zmath  adsnasa  isi  scopus
    22. Krattenthaler, C, “Hypergeometry and Riemann's zeta function”, Memoirs of the American Mathematical Society, 186:875 (2007), VII  crossref  mathscinet  isi
    23. Cresson, PJ, “Multiple hypergeometric series and polyzetas”, Bulletin de La Societe Mathematique de France, 136:1 (2008), 97  crossref  mathscinet  zmath  isi  scopus
    24. Pilehrood, KH, “An Apery-like continued fraction for pi coth pi”, Journal of Difference Equations and Applications, 14:12 (2008), 1279  crossref  mathscinet  zmath  isi  scopus
    25. Nesterenko Yu.V., “Construction of Approximations to Zeta-Values”, Diophantine Approximation: Festschrift for Wolfgang Schmidt, Developments in Mathematics, 16, eds. Schlickewei H., Schmidt K., Tichy R., Springer, 2008, 275–293  crossref  mathscinet  zmath  isi  scopus  scopus
    26. Rivoal, T, “Arithmetic applications of Lagrangian interpolation”, International Journal of Number Theory, 5:2 (2009), 185  crossref  mathscinet  zmath  isi  scopus
    27. Sondow, J, “A hypergeometric approach, via linear forms involving logarithms, to criteria for irrationality of Euler's constant”, Mathematica Slovaca, 59:3 (2009), 307  crossref  mathscinet  zmath  isi  elib  scopus
    28. Fischler S., “Restricted rational approximation and Apery-type constructions”, Indagationes Mathematicae-New Series, 20:2 (2009), 201–215  crossref  mathscinet  zmath  isi  scopus  scopus
    29. Gutnik L., “Elementary Proof of Yu. V. Nesterenko Expansion of the Number Zeta(3) in Continued Fraction”, Advances in Difference Equations, 2010, 143521  crossref  mathscinet  zmath  isi
    30. Yu. V. Nesterenko, “On the Irrationality Exponent of the Number $\ln2$”, Math. Notes, 88:4 (2010), 530–543  mathnet  crossref  crossref  mathscinet  isi
    31. T. Rivoal, “Linear forms in zeta values arising from certain Sorokin-type integrals”, J. Math. Sci., 180:5 (2012), 641–649  mathnet  crossref  mathscinet  elib
    32. W. Zudilin, “Arithmetic hypergeometric series”, Russian Math. Surveys, 66:2 (2011), 369–420  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    33. Huttner M., “Riemann Beta-Scheme, Monodromy and Diophantine Approximations”, Indag. Math.-New Ser., 23:3 (2012), 522–546  crossref  mathscinet  zmath  isi  scopus  scopus
    34. Mortici C., “Estimating the Apery's Constant”, J. Comput. Anal. Appl., 14:2 (2012), 278–282  mathscinet  zmath  isi
    35. A. A. Polyanskii, “Square exponent of irrationality of $\ln 2$”, Moscow University Mathematics Bulletin, 67:1 (2012), 23–28  mathnet  crossref
    36. Pilehrood Kh.H. Pilehrood T.H., “On a Continued Fraction Expansion for Euler's Constant”, J. Number Theory, 133:2 (2013), 769–786  crossref  mathscinet  zmath  isi  scopus  scopus
    37. Lagarias J.C., “Euler's Constant: Euler's Work and Modern Developments”, Bull. Amer. Math. Soc., 50:4 (2013), 527–628  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    38. E. A. Karatsuba, “On One method for constructing a family of approximations of zeta constants by rational fractions”, Problems Inform. Transmission, 51:4 (2015), 378–390  mathnet  crossref  isi  elib
    39. Soria Lorente A., “On Zudilin-Like Rational Approximations to ? (5)”, Notes Number Theory Discret. Math., 24:2 (2018), 104–116  crossref  isi
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